Abstract

This thesis focuses on the stability of spatial localized solutions of lattice dynamical systems (LDSs). In particular, I focus on the stability of spatially localized single- and multi-pulse solutions to lattice dynamical systems. By linearizing the nonlinear system around steady-state solutions and applying exponential dichotomy theory, an isomorphism between the localized solutions and the stable front and back solutions is constructed. Then, the proof constructs Evans functions about the localized solutions from Evans functions of the front and back solutions. It gives that eigenvalues of the front and back solutions lead to nearby eigenvalues for the localized solution, which can in turn be used to verify instability of the solution. Furthermore, Rouché's theorem is used to prove that the number of roots in the single-pulse solution is the sum of front and back solutions', and the multi-pulse solution's is the sum of each single-pulse solution's. This work is widely applicable to a range of scalar LDSs, particularly the well-studied discrete Nagumo equation. Finally, this thesis concludes with a discussion of possible avenues for future work.